I'm going to attempt a different answer, based on what I think you're trying to get at in the comments, especially since you've been asking how things look when one simulates field theory on a lattice.
So let's set up our theory with an explicit cutoff, $\Lambda > 0$, which can be a placeholder for anything you wish (an inverse lattice spacing, a hard momentum cutoff, etc.). We'll just take Euclidean $\phi^4$ theory:
$$
\mathcal{Z} = \int_{\Lambda} \mathcal{D} \phi \exp\left[ - \int d^2 x \left( \frac{1}{2} \left( \nabla \phi \right)^2 + \frac{m^2}{2} \phi^2 + \frac{\lambda}{4!} \phi^4 \right) \right].
$$
In what follows, we will never completely drop the bare parameters $m^2$, $\lambda > 0$, but I will introduce particular observables and renormalized couplings. This way, we are being very careful in taking the $\Lambda \rightarrow \infty$ limit (which formally causes the bare couplings to diverge), which is helpful because this limit is secretly the scaling limit. The scaling limit everywhere away from the critical point is a free Gaussian, where the fields then have trivial scaling dimensions. I think this last fact is what may be causing your confusion.
First, to answer your question:
Concretely what this should mean in terms of a lattice simulation is the following. If we change the lattice spacing, using a finer or coarser grid, we can get the exact same correlation functions as before (at scales much larger than the lattice spacing). All we have to do is change the bare mass parameter in the Lagrangian. The 𝜆
λ
parameter and the field normalization can stay exactly the same.
Is this non-perturbative statement about correlation functions on the lattice correct? This is a very concrete question which should be able to answered by someone who has done lattice simulations.
The answer to the question is no: for a particular fine-tuned value of $m_c^2$, whose exact value depends on your choice of $\lambda$ and $\Lambda$, this theory "flows" to the Ising CFT in the scaling limit. You will likely be interested in the following reference which studies this theory numerically: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.91.085011. (I admit I haven't read it since it came out in 2015 and I was a much weaker physicist.) The approach to the scaling limit can be thought of in two different ways:
- Take $\Lambda$ fixed, but consider studying our theory on very long length scales. If we compute the two-point function at the critical point, we find
$$
\lim_{x \rightarrow \infty} \Gamma^{(2)}(x)^{-1} = \lim_{x \rightarrow \infty} \langle \phi(x) \phi(0) \rangle \propto \frac{1}{|\Lambda x|^{1/4}}.
$$
with some dimensionless non-universal coefficient. If we deviate the coupling $m^2$ a small amount from the critical point, we find that this correlator asymptotically decays as $e^{-|x|/\xi}$, where $\xi \propto (m^2 - m_c^2)^{-\nu}$ where $\nu = 1$ in our case.
- Consider a family of theories which take varying values of $m^2$, $\lambda$, and $\Lambda$. Now compute exactly the observables like $\Gamma^{(n)}$ and $\xi$ as analytic functions of $m^2$, $\lambda$, and $\Lambda$ (easier said than done, I know). So you have access to the exact functions
$$
\Gamma^{(n)}(x) = \Gamma^{(n)}(x,m^2,\lambda,\Lambda), \qquad \xi = \xi(m^2,\lambda,\Lambda).
$$
Now it turns out that there is a precise limit of the form $\Lambda \rightarrow \infty$, $m^2 \rightarrow m_c^2$ ($m_c$ is non-universal, and also generically divergent in this process), such that $\xi \rightarrow \infty$ and the $\Gamma^{(n)}$'s take the functional form of the Ising CFT (with appropriate overall factors of $\Lambda$ for dimensional analysis). A rather large complication here involves finding the right way to take $m^2 \rightarrow m_c^2$: for a critical theory with critical exponent $\nu$, one needs the dependence
$$
\xi \sim (m^2 - m_c^2)^{-\nu}.
$$
The usual procedure for dealing with this is to systematically (within perturbation theory) eliminate the bare quantities $m^2$, $\lambda$, in terms of interesting "observable" quantities like $\xi$ or $\lambda_R(k) = \Gamma^{(4)}(k)$, and then the limit of $\Lambda \rightarrow \infty$ which leaves $\xi$ in the scaling regime is less opaque. This whole procedure is way messier conceptually than the first, but it's actually conducive to analytic calculations.
These two pictures are equivalent because in both procedures you're taking the ratio of all large length scales ($\xi$ and $x$) and your one small length scale ($\Lambda^{-1}$) to infinity.
This is not in conflict with perturbation theory. Let's now choose $\Lambda$ to be a hard cutoff in momentum space, so all momenta satisfy $k^2 \equiv |\mathbf{k}|^2 \leq \Lambda^2$. Now we can choose to compute the correlation length, which is given by the inverse two-point function evaluated at $k^2 = 0$. I find
$$
\xi^{-2} \equiv \Gamma^{(2)}(k = 0) = m^2 + \frac{\lambda}{8 \pi} \log \left[ 1 + \frac{\Lambda^2}{m^2} \right] + \frac{\lambda^2}{64 \pi^2 m^2 (m^2 + \Lambda^2)}\log \left[ 1 + \frac{\Lambda^2}{m^2} \right] + \frac{\lambda^2}{m^4} F(m^2/\Lambda^2) + \cdots.
\tag{*}\label{*}
$$
Here I have neglected to explicitly compute the finite sunset diagram, represented here by $F(m^2/\Lambda^2)$, but the terms shown will suffice to show the main point.
In a QFT course, you now introduce a renormalized mass,
$$
m_R^2 = m^2 - \frac{\lambda}{8 \pi} \log \left[ 1 + \frac{\Lambda^2}{m^2} \right],
$$
and now the theory is completely finite in the limit $\Lambda \rightarrow \infty$; explicitly
$$
\xi^{-2} = m_R^2 + \frac{\lambda^3}{m_R^4} F(0).
$$
But this is only part of the scaling limit! One also wants to take $\xi \rightarrow \infty$, which involves taking $m_R \rightarrow 0$ at this order in perturbation theory. Returning back to Eq. \eqref{*}, one sees that in perturbation theory with a finite cutoff, non-analytic contributions to $\xi$ occur in both the $\Lambda \rightarrow \infty$ and $m^2 \rightarrow 0$ limits, and while we can deal with the dependence on $\Lambda$ to all orders with renormalization, we are not taking into account the non-analytic behavior due to the $m^2 \rightarrow m_c^2$ limit which appears at every order. This is unacceptable, because as outlined above, correctly obtaining the dependence of physical observables on $(m^2 - m_c^2)$ is crucial for taking the scaling limit.
The way that the famous Wilson-Fisher $\epsilon = 4 - d$ expansion gets around this is that the IR and UV divergences are both logarithmic just under the upper-critical dimension, resulting in terms like $\log(m/\Lambda)$. Then the Callan-Symanzik equations can be used to resum the logarithms into bonafide power laws (the coefficients of the logs become related to critical exponents). But in the $d=2$ case, one either works with a massive theory, where the early $\Lambda \rightarrow \infty$ limit shoots us deep into the adjacent phases, or one tries to work with the massless theory, which is plagued by incurable IR divergences.
